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Chapter 10: Problem 5

If Laura invests \(\$ 100\) at \(6 \%\) interest compounded quarterly, how many months must she wait for her money to double? (She cannot withdraw the money before the quarter is up.)

Short Answer

Expert verified
To calculate 't', use the values and the constant 2 representing the conversion factor for years to months: 't' = ln(2) / (4 * ln(1 + 0.06/4)) * 12 (Put ln(2), ln(1 + 0.06/4), and 12 into a calculator to get the exact value).

Step by step solution

01

Identify the known quantities

The known quantities in this problem are: Initial principal (\$100), Interest rate (6% per annum already divided by 100 to convert it into a decimal, which makes it 0.06), Period of investment (unknown, this is what we need to find), and Since the interest is compounded quarterly, the number of compounding periods per year (n) is 4.
02

Set Up the Equation

We know that the final amount (A) is double the initial principal, which means A = 2 * P. The formula for compound interest is A = P(1 + r/n)^(nt). Substituting the known values into the equation gives us 2*P = P(1 + 0.06/4)^(4t). Since we are interested in getting the number of quarters (4t), we need to isolate 4t on one side of the equation.
03

Solve the Equation

To this end, divide both sides of the equation by P, and you get 2 = (1 + 0.06/4)^(4t). Then, take the natural log (ln) of both sides to get rid of the exponent: ln(2) = ln((1 + 0.06/4)^(4t)) = 4t * ln(1 + 0.06/4). Finally to get 't', divide both sides by 4 * ln(1 + 0.06/4) which gives t = ln(2) / (4 * ln(1 + 0.06/4)).
04

Convert 't' into Months

The value of 't' obtained will be in years. To convert it into months, multiply 't' by 12.

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